convex_le — Mathlib

English

The convexity of the set uIcc(r,s) is ensured in linear ordered contexts; it matches convex_Icc when appropriate.

Русский

Выпуклость $k$-множество $uIcc(r,s)$ обеспечена в линейно упорядоченном контексте.

LaTeX

$$$\operatorname{Convex}_\mathbb{K}(\{x : r \le x \le s\})$$$

Lean4

theorem convex_le (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β) : Convex 𝕜 ({x ∈ s | f x ≤ r}) :=
  fun x hx y hy _ _ ha hb hab =>
  ⟨hs hx.1 hy.1 ha hb hab,
    (hf (hs hx.1 hy.1 ha hb hab) (max_rec' s hx.1 hy.1) (Convex.combo_le_max x y ha hb hab)).trans
      (max_rec' {x | f x ≤ r} hx.2 hy.2)⟩

Похожие утверждения